A024924 a(n) = Sum_{k=1..n} prime(k)*floor(n/prime(k)).

0, 0, 2, 5, 7, 12, 17, 24, 26, 29, 36, 47, 52, 65, 74, 82, 84, 101, 106, 125, 132, 142, 155, 178, 183, 188, 203, 206, 215, 244, 254, 285, 287, 301, 320, 332, 337, 374, 395, 411, 418, 459, 471, 514, 527, 535, 560, 607, 612, 619, 626, 646, 661, 714, 719, 735, 744, 766, 797, 856

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

For n > 2, sum of all distinct prime factors composing numbers from 2 to n.

References

• M. Kalecki, On certain sums extended over primes or prime factors (in Polish), Prace Mat., Vol. 8 (1963/64), pp. 121-129.
• József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter IV, p. 144.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Vaclav Kotesovec, Plot of a(n) / ((Pi^2/12) * n^2/log(n)) for n = 1..10^8

Crossrefs

Partial sums of A008472.

Cf. A000720, A024934, A034387.

Programs

Magma

[0] cat [ &+[ NthPrime(k)*Floor(n/NthPrime(k)): k in [1..n]]: n in [1..60]]; // Vincenzo Librandi, Jul 28 2019

Mathematica

Join[{0}, Table[Sum[Prime[k] Floor[n / Prime[k]], {k, 1, n}], {n, 1, 60}]] (* Vincenzo Librandi, Jul 28 2019 *)
Join[{0}, Accumulate[Table[Sum[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}]]] (* Vaclav Kotesovec, May 20 2020 *)

PARI

a(n) = sum(k=1, n, prime(k)*(n\prime(k))); \\ Michel Marcus, Mar 01 2015

PARI

a(n)=my(s); forprime(p=2,n, s+=n\p*p); s \\ Charles R Greathouse IV, Jun 26 2020

Python

from sympy import prime
def A024924(n): return sum((p:=prime(k))*(n//p) for k in range(1,n+1)) # Chai Wah Wu, Sep 18 2023

Formula

a(n) = n*A000720(n) - A024934(n). - Max Alekseyev, Feb 10 2012

a(n) = A034387([n/1]) + A034387([n/2]) + ... + A034387([n/n]). Terms can be computed efficiently with the following formula: a(n) = A034387([n/1]) + ... + A034387([n/m]) - m*A034387([n/m]) + Sum_{prime p<=n/m} p*[n/p], where m = [sqrt(n)]. - Max Alekseyev, Feb 10 2012

G.f.: Sum_{k >=1} (prime(k)*x^prime(k)/(1-x^prime(k)))/(1-x). - Vladeta Jovovic, Aug 11 2004

a(n) ~ ((Pi^2 + o(1))/12) * n^2/log(n) (Kalecki, 1963/64). - Amiram Eldar, Mar 04 2021

Extensions

a(0)=0 prepended by Max Alekseyev, Feb 10 2012